The first question, on entering the completed interior of the church of Sagrada Família, is: "Is it really there?" We have been so long accustomed to the idea that Barcelona's most famous landmark is a permanent ruin, unfinished and unfinishable, that it comes as a shock to find it is now keeping out the weather, for the first time in its 130 years of making. The canned ecstasy of the Hallelujah Chorus plays on the PA system and sunbeams pierce its forest of columns with such dazzle and precision that, you think, they must be digital. It is like walking into the Colosseum and finding it all there, with awnings, crowds, sand, blood, beasts, gladiators and thumb-turning emperor which, being clearly impossible, would most easily be explained as a video game in three dimensions.

The second question is: "Is it really Gaudí?" The great Catalan architect famously adjusted his buildings as he went along, modifying details in response to unusual stones found in the quarry and forever testing his ideas with full size mock-ups. He had a donkey hoisted up the facade of the church, to see how it would look in a sculpted nativity scene, and made plaster casts of temporarily anaesthetised turkeys and chickens and, so he could model a Massacre of the Innocents, of stillborn babies. In the interests of spiritual research, he attended a death at a hospital and claimed he could see the moment when the soul of the departed met the holy family. Gaudí was fatally hit by a tram in 1926 and no subsequent architect working on the church has come close to matching his fanaticism or genius.

True, he left large plaster models of the nave, big enough to walk through, and of key elements. He left somewhat blurry drawings of the whole, including an overwhelming 170-metre cucumber of a tower, which is yet to be built. But these models and drawings leave much undefined and, as Gaudí himself changed his mind during the development of the church, it seems likely that he would have continued to do so had he overseen its completion.

According to Oriol Bohigas, the octogenarian architect who oversaw Barcelona's remaking of itself from the 1980s on, the completion of the church makes it, architecturally speaking at least, into "the most reactionary city in Europe". His business partner, the British-born David Mackay, elaborates: "It's doubtful whether you can continue the work after such a long time and claim it's Gaudí's building." It is at best "an interpretation" or a "full-size version of the model".

Jordi Bonet, another octogenarian and architect of the building work since 1985, disagrees. "Gaudí's wishes are very clear: to continue the building of the basilica," he says. "This is being undertaken with the utmost fidelity to his ideas. He always spoke of his successors, giving them the necessary interpretative licence. The naves, the roofs, the columns, the ceiling vaults are exactly as he modelled them and follow the geometrical and structural rules that Gaudí set up, allowing us to build exactly as he set the project out."

Pope Benedict XVI celebrates Mass and consecrates the Sagrada Família in November last year during a visit to Spain. Christophe Simon/AFP/Getty Images

The debate has been given added force by the completion of the nave last autumn, by its consecration by the Pope and by the recent decision of the city of Barcelona to award the new work its highest architectural prize, but it goes back decades. In the early 1960s, architectural luminaries such as Le Corbusier and Alvar Aalto signed a petition, urging that the church either be left unfinished or that a competition be held to find a new design by a living architect. Oscar Tusquets Blanca, who became a leading Barcelona architect, helped organise the petition as a student. He now says that its main effect was to prompt a record-breaking year for public donations to the building effort, in reaction to this intervention by "Marxist heretics".

The argument is not only about architecture, but also about religion, and it goes back to Josep María Bocabella, the devout and eccentric bookseller who first conceived the idea of building a great church. It was to be an affirmation of the Catholic church, in the face of threats from a secular industrial society. The church would be dedicated to the Holy Family, in order to buttress family life, and would be placed on the edge of the expanding city. Early photographs show flocks of goats being herded in front of the building site. Construction started in 1882 and there were hopes it would be ready for use within a decade.

After parting company with his first architect, Bocabella appointed the 31-year-old Gaudí. According to legend, he dreamed that his architect would have piercing blue eyes and then met Gaudí, who had such eyes. It is possible that he thought he was getting a cheap option, as the young man would have charged lower fees than more established competitors. If so, this hope was as vain as the projected timetable. Whatever might have been saved on fees was spent many times over on Gaudí's ambitious design.

Time and budget are usually the main constraints on building projects, but here both counted for nothing. It is impossible to know how much the church has cost so far, and will cost to finish, and no one has ever known how long it will take. "My client," said Gaudí, meaning God, "is not in a hurry." What mattered was how truly his vision of the church would reflect its spiritual ambitions and if he got this right the funds would follow – from a shop that donated a peseta a day, from larger donors offering indulgences and papal blessings, and from special fundraising days. "In the Sagrada Família, everything is providential," said Gaudí.

Gaudí's career was established with the Sagrada Família and it ended with it, when, out of fashion, he had almost no other work left. In between, he designed other works that show more dazzling invention, more freedom of imagination and more joy than the sometimes lugubrious gothic of the church, but for him it was his most important work. If he was not, after all, a cheap option, there was no doubting his suitability on the grounds of faith. He was exceptionally devout, once going without food for the 40 days of Lent, in emulation of Christ.

As Gijs van Hensbergen's fine biography of Gaudí records, his ascetic existence included breakfasts of burned toast, and lunches of lettuce leaves dipped in milk. His old suits would be tinged with green mould, he wore shoes made of courgette roots and after his collision with the tram he was found to be wearing underpants held together by safety pins. He believed that Catalonia had been chosen by God to take forward Christian architecture. He also had the implacable self-belief of an artist. "You've either got to kill him," said a contemporary, "or give in and tell him he's right."

His building is dense with his fervour. It strives to compress all of earth and heaven into its structure – endless saints, biblical scenes, symbols, inscriptions, seashells, reptiles, birds, flowers and fruit. Time was captured through images of the seasons and holy dates. It was not just a thing of sight – the spires are designed for peals of bells, the nave for a choir of up to 1,500.

Cranes crowd around the basilica’s high towers. Estimates place completion of the project some time between 10 and 15 years from now. Photograph: Samuel Aranda/Getty Images

With its avoidance of straight lines and right angles, and its tree-like columns, it embodies Gaudí's belief that he should follow nature. Above all, it has the property of fusion: on the Facade of the Nativity, the most significant part built in Gaudí's lifetime, columns and arches melt into a viscous jism that foams, drips and procreates foliage, beasts and people. It then becomes the geological eruption that is the building itself, in whose spires and portals you can, without difficulty and should you wish, read further sexual images. No other architect has made stone look so fluid, so dissolving. It is not pretty, but that is not the point. As Salvador Dalí, an early fan, put it: "Those who have not tasted his superbly creative bad taste are traitors."

After Gaudí's death, construction inched forward, until the world's discovery of the Catalan city as dream tourist destination, as celebrated in Freddie Mercury's kitschy anthem "Barcelona", in the years leading up to the 1992 Olympics. Gaudí's buildings were central to this attraction and paying visitors started turning up at the church in ever-increasing numbers. Now, more than 2 million people a year pay €12.50 a time to see the church, a never-ending source of income that makes construction unstoppable. It has become a gawp factory, a perpetual motion machine in which tourist income feeds construction which feeds more tourism.

The Passion Facade, a grim counterpart to the lush Nativity Facade, was built. It has sculptures of cartoonish anguish by artist Josep María Subirachs, the awfulness of which is beyond description. Next came the nave, which follows the model exactly, but adds ornaments such as glistening medallions of the evangelists, which look like gift shop tat. The vast tower and the main entrance – the Facade of the Glory – are expected to be finished some time in the next decade.

What is there now is the progeny of the strange coupling of Freddie Mercury and the Holy See, of mass tourism and faith. It offers religion as spectacle; with a capacity approaching 15,000, it is not far short of event venues like the O2. As far as its builders are concerned, they are fulfilling the dream of the bookseller Bocabella, which was to build a church. It would be a nonsense, for them, to leave the job unfinished. If the building is monstrously, outrageously kitsch – and it is – it bothers neither them, nor the flocks of visitors.

What is there now is impressive for its scale, complexity and persistence. Oscar Tusquets Blanca, an old enemy of the continued building programe, has been converted. "It is incredible… something unique. It is a pity the details are wrong but if they were to give me €3m and six months, I could solve the problem." That it is also ugly, in places repulsive, is, as Dalí observed, part of the point.

But it is no longer a work of Gaudí. It cannot overcome the central paradox, which is that Gaudí's architecture was organic, living and responsive, whereas posthumous simulation of his ideas makes them fixed and lifeless. The fusion, the melting, the integration of structure and ornament and the demented frenzy that drove Gaudí to do strange things with comatose turkeys and dead babies cannot be replicated.

What is to be done? It is not possible to return to the romantic ruin it was once. In all seriousness, with due regard to health and safety, and in no way condoning the arson attack that was made on the church last week, I would suggest distressing the new work with machine guns. Not only would this reduce the offensiveness of the terrible sculptures, but it would also erode that computerised precision, that deadliness, that lack of Gaudí's solubility, which is the worst feature of the new work.